|
25
|
1 |
(* Title: Adaptation of example from HOL/Hoare/Separation
|
|
|
2 |
Author: Rafal Kolanski, 2012
|
|
|
3 |
Maintainers: Gerwin Klein <kleing at cse.unsw.edu.au>
|
|
|
4 |
Rafal Kolanski <rafal.kolanski at nicta.com.au>
|
|
|
5 |
*)
|
|
|
6 |
|
|
|
7 |
header "Separation Algebra for Virtual Memory"
|
|
|
8 |
|
|
|
9 |
theory VM_Example
|
|
|
10 |
imports "../Sep_Tactics" "../Map_Extra"
|
|
|
11 |
begin
|
|
|
12 |
|
|
|
13 |
text {*
|
|
|
14 |
Example instantiation of the abstract separation algebra to the sliced-memory
|
|
|
15 |
model used for building a separation logic in ``Verification of Programs in
|
|
|
16 |
Virtual Memory Using Separation Logic'' (PhD Thesis) by Rafal Kolanski.
|
|
|
17 |
|
|
|
18 |
We wrap up the concept of physical and virtual pointers as well as value
|
|
|
19 |
(usually a byte), and the page table root, into a datatype for instantiation.
|
|
|
20 |
This avoids having to produce a hierarchy of type classes.
|
|
|
21 |
|
|
|
22 |
The result is more general than the original. It does not mention the types
|
|
|
23 |
of pointers or virtual memory addresses. Instead of supporting only
|
|
|
24 |
singleton page table roots, we now support sets so we can identify a single
|
|
|
25 |
0 for the monoid.
|
|
|
26 |
This models multiple page tables in memory, whereas the original logic was
|
|
|
27 |
only capable of one at a time.
|
|
|
28 |
*}
|
|
|
29 |
|
|
|
30 |
datatype ('p,'v,'value,'r) vm_sep_state
|
|
|
31 |
= VMSepState "((('p \<times> 'v) \<rightharpoonup> 'value) \<times> 'r set)"
|
|
|
32 |
|
|
|
33 |
instantiation vm_sep_state :: (type, type, type, type) sep_algebra
|
|
|
34 |
begin
|
|
|
35 |
|
|
|
36 |
fun
|
|
|
37 |
vm_heap :: "('a,'b,'c,'d) vm_sep_state \<Rightarrow> (('a \<times> 'b) \<rightharpoonup> 'c)" where
|
|
|
38 |
"vm_heap (VMSepState (h,r)) = h"
|
|
|
39 |
|
|
|
40 |
fun
|
|
|
41 |
vm_root :: "('a,'b,'c,'d) vm_sep_state \<Rightarrow> 'd set" where
|
|
|
42 |
"vm_root (VMSepState (h,r)) = r"
|
|
|
43 |
|
|
|
44 |
definition
|
|
|
45 |
sep_disj_vm_sep_state :: "('a, 'b, 'c, 'd) vm_sep_state
|
|
|
46 |
\<Rightarrow> ('a, 'b, 'c, 'd) vm_sep_state \<Rightarrow> bool" where
|
|
|
47 |
"sep_disj_vm_sep_state x y = vm_heap x \<bottom> vm_heap y"
|
|
|
48 |
|
|
|
49 |
definition
|
|
|
50 |
zero_vm_sep_state :: "('a, 'b, 'c, 'd) vm_sep_state" where
|
|
|
51 |
"zero_vm_sep_state \<equiv> VMSepState (empty, {})"
|
|
|
52 |
|
|
|
53 |
fun
|
|
|
54 |
plus_vm_sep_state :: "('a, 'b, 'c, 'd) vm_sep_state
|
|
|
55 |
\<Rightarrow> ('a, 'b, 'c, 'd) vm_sep_state
|
|
|
56 |
\<Rightarrow> ('a, 'b, 'c, 'd) vm_sep_state" where
|
|
|
57 |
"plus_vm_sep_state (VMSepState (x,r)) (VMSepState (y,r'))
|
|
|
58 |
= VMSepState (x ++ y, r \<union> r')"
|
|
|
59 |
|
|
|
60 |
instance
|
|
|
61 |
apply default
|
|
|
62 |
apply (simp add: zero_vm_sep_state_def sep_disj_vm_sep_state_def)
|
|
|
63 |
apply (fastforce simp: sep_disj_vm_sep_state_def map_disj_def)
|
|
|
64 |
apply (case_tac x, clarsimp simp: zero_vm_sep_state_def)
|
|
|
65 |
apply (case_tac x, case_tac y)
|
|
|
66 |
apply (fastforce simp: sep_disj_vm_sep_state_def map_add_ac)
|
|
|
67 |
apply (case_tac x, case_tac y, case_tac z)
|
|
|
68 |
apply (fastforce simp: sep_disj_vm_sep_state_def)
|
|
|
69 |
apply (case_tac x, case_tac y, case_tac z)
|
|
|
70 |
apply (fastforce simp: sep_disj_vm_sep_state_def map_add_disj)
|
|
|
71 |
apply (case_tac x, case_tac y, case_tac z)
|
|
|
72 |
apply (fastforce simp: sep_disj_vm_sep_state_def map_add_disj map_disj_com)
|
|
|
73 |
done
|
|
|
74 |
|
|
|
75 |
end
|
|
|
76 |
|
|
|
77 |
end
|
|
|
78 |
|